At this stage, many students raise the following questions, which turn out to be related to one another:
According to Einstein, if observers A and B aren’t at rest relative to each other, then A says B’s time is slow, but B says A is the slow one. How can this be? If A says B is slow, shouldn’t B say A is fast? After all, if I took a pill that sped up my brain, everyone else would seem slow to me, and I would seem fast to them.
Suppose I keep accelerating my spaceship steadily. What happens when I get to the speed of light?
In all the diagrams in Section 1.4, the parallelograms have their diagonals stretched and squished by a certain factor, which depends on v. What is the interpretation of this factor?
The stretching and squishing factors for the diagonals are the same as the Doppler shift. We notate this factor as D (which can stand for either “Doppler” or “diagonal”).
In nonrelativistic physics, velocities add in relative motion. For example, if a boat moves relative to a river, and the river moves relative to the land, then the boat’s velocity relative to the land is found by vector addition. This linear behavior cannot hold relativistically.
No continuous process of acceleration can boost a material object to c . That is, the subluminal (slower than light) nature of a electron or a person is a fundamental feature of its identity and can never be changed. Einstein can never get on his motorcycle and drive at c as he imagined when he was a young man, so we material beings can never see the world from a frame of reference that travels at c.
A frequent source of confusion in relativity is that we write down equations that are coordinate-dependent, but forget the dependency. Similarly, it is possible to write expressions that are only valid for one choice of signature. The following notation, defining a projection operator P , is one tool for avoiding these difficulties.
Special relativity doesn’t permit the existence of observers who move at c . This is because if two observers differ in velocity by c , then the Lorentz transformation between them is not a one-to-one map, which is physically unacceptable.
